Solutions of Alkylammonium and Bulky Anions: Description of Osmotic

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Solutions of Alkylammonium and Bulky Anions: Description of Osmotic Coefficients within the Binding Mean Spherical Approximation Nicolas Papaiconomou,*,† Jean-Pierre Simonin,‡ and Olivier Bernard‡ †

LCME, Université de Savoie, Bâtiment Chartreuse, Savoie Technolac, 73376 Le Bourget du Lac CEDEX, France Laboratoire PECSA (UMR CNRS 7195), Université Pierre et Marie Curie, Case 51, 75252 Paris Cedex 05, France ‡

S Supporting Information *

ABSTRACT: The binding mean spherical approximation (BiMSA) is used to describe osmotic coefficients for aqueous solutions of salts containing alkylammonium cations or bulky anions. A total of 35 salt solutions is accurately described at 25 °C over the whole concentration range, up to very high concentrations such as 20 mol·kg−1 for methylammonium chloride or 24 mol·kg−1 for ammonium thiocyanate. The ion diameters, the permittivity of solution, and the association constant are adjustable parameters within the BiMSA model. New diameter values are assigned to alkylammonium cations and bulky anions such as tetrafluoroborate, alkanesulfonates, methylsulfate, trifluoroacetate, or trifluoromethanesulfonate anions. Alkylammonium sizes are in reasonable agreement with literature values. Besides, association constants values obtained within the BiMSA model compare well with literature values when available. when butyl chains are appended to a nitrogen atom,11 solutions of tetraalkylammonium salts containing butyl or longer alkyl chains are not described in this work. Furthermore, in order to test the ability of the model to describe salts containing asymmetrical ions, we describe the properties of salts based on polymethylammonium cations and the chloride, nitrate, or perchlorate anion and a dibolaform cation ethylenebis(trimethylammonium) with the iodide, nitrate, or sulfate anion. These salts will be referred to as set S1. The anions within set S1 have already been described within the BiMSA model.4,5 The diameters previously reported are used in this work. A second set of solutions (set S2), containing the H+, Li+, or Na+ cations combined with the bulky tetrafluoroborate, acetate, methylsulfate, or alkanesulfonate anion that usually compose ionic liquids, are also described within the BiMSA model. In this set, the diameters of the cations at infinite dilution are taken from previous work.4 Because experimental values for the osmotic or mean ionic activity coefficients are reported on the molality scale, as opposed to those calculated on the molarity scale within the BiMSA model, density values are required. The latter are available in the literature for most of the salts studied here.35−47 However, density data for aqueous solutions containing the alkanesulfonate anion, that could not be found in the literature, have been measured in this study for solutions of HCH3SO3, NaCH3SO3, HC2H5SO3, and NaC2H5SO3 at 25 °C and up to high concentrations.

1. INTRODUCTION The mean spherical approximation (MSA)1−4 is a statistical thermodynamic model for the representation of the thermodynamic properties of electrolyte solutions, regarding an ion i as a charged sphere of diameter σi immersed in a solvent of permittivity ε. The MSA model uses parameters (ion sizes and relative solution permittivity) that have some physical meaning. For applications to real solutions, crystallographic values are taken for the halide anions radii and the size of a given cation at infinite dilution is the same for all salts containing this cation.5 Within the binding MSA (BiMSA) version of the model,6,7 an effective association constant is introduced, taking into account the pairing or association of unlike ions. This model has been shown to provide good description for deviations from ideality in aqueous solutions of salts such as alkali halides, nitrates, perchlorates, sulfates,6−8 or, more recently, lanthanide salts9,10 or mixtures of salts.5,7,10 The BiMSA model has been widely used to describe monatomic cations. An attempt at describing salts containing a tetramethyl-, tetraethyl-, or tetrapropylammonium cation and a halide anion has previously been reported.6 Nevertheless, the results obtained at the time are not satisfying, because despite an accurate description of the osmotic coefficients of tetraalkylammonium halide solutions, the values for the adjusted parameters, such as the cation diameter, were not plausible. Therefore, in order to extend the number of solutions described within the BiMSA model and in order to describe solutions containing large and polyatomic cations or anions, we apply here the BiMSA model to the description of aqueous solutions containing tetraalkylammonium salts with methyl, ethyl, or propyl alkyl chains and with the chloride, bromide, iodide, or nitrate anion. Because micelles might be formed © 2012 American Chemical Society

Received: Revised: Accepted: Published: 9661

December 16, 2011 May 25, 2012 June 25, 2012 June 25, 2012 dx.doi.org/10.1021/ie202954y | Ind. Eng. Chem. Res. 2012, 51, 9661−9668

Industrial & Engineering Chemistry Research

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This work will therefore be divided in the following sections. First, the experimental method used for measuring density will be exposed. Then, a short summary of the BiMSA model used in this work will be given. The density values of all solutions studied here will then be presented. The results and discussion section devoted to the description of osmotic coefficients will be divided in three subsections, depending on the nature of the ions described. The case of tetraalkylammonium solutions will be considered first, followed by polymethylammonium solutions. Finally, the description of solutions containing bulky anions (set S2) will be presented and discussed. It will be followed by a short conclusion.

with ΔABiMSA = ΔAel + ΔAMAL

We consider a two-component electrolyte in which only one cation and one anion can form a pair. A pair is defined as a cation and an anion being in contact. K

A+ + B− ↔ AB

with K as the equilibrium constant of association. The total (free and associated) number densities of cation and anion are ρ+ and ρ−, respectively. We denote by α+ and α− the unbound ion fractions. The mass action law (MAL) is then written as9

2. EXPERIMENTAL SECTION The densities of methanesulfonic acid (HCH3SO3) and ethanesulfonic acid (HC2H5SO3), sodium methanesulfonate (NaCH3SO3), sodium ethanesulfonate (NaC2H5SO3), and trimethylammonium chloride (NMe3HCl) were measured in this work. Concentrated solutions of HCH 3 SO 3 and HC2H5SO3 were purchased from Sigma Aldrich and used without further purification. Titration of the acidic solutions was carried out with a commercial solution of 1 M NaOH. The resulting concentrations of the HCH3SO3 and HC2H5SO3 solutions were found to be 9.52 and 7.96 mol·L−1, respectively, in agreement with the manufacturer specifications. Diluted solutions with concentrations ranging from 0.1 to 9.52 or 7.96 mol·L−1, accordingly, were then prepared and their densities were measured using a 5 mL pycnometer. Next, neutralization of the methanesulfonate and ethanesulfonate acid solutions was carried out with a 1 M NaOH solution. This yielded aqueous solutions of sodium methanesulfonate and sodium ethanesulfonate, respectively. Again, the densities were measured with a 5 mL pycnometer. A set of density data was obtained by diluting the stock solution of sodium alkanesulfonate. Concentrations for the solutions were in the range 0.05−1 mol·L−1. NMe3HCl was purchased from Alfa Aesar and used as received. Diluted solutions with concentrations ranging from 0.1 to 2.5 mol·kg−1 were prepared, and their densities were measured with a 5 mL pycnometer. The precision of the pycnometer was ±0.4%. Our measurements were validated with a commercially available solution of HCF3CO2 titrated with a 1 M NaOH solution. Densities for the resulting NaCF3CO2 aqueous solutions at concentrations ranging from 0.2 to 2.8 mol·kg−1 were compared to literature values.12 The deviation between our measurements and those previously reported was found to be in all cases below 1%.

K0 ≡

⎡ z′ z′ z z ⎤ = KgcHS exp⎢ −2λ + − + 2λ 0 + − ⎥ ⎢⎣ σP σP,0 ⎥⎦ (ρ+ α+)(ρ− α −) ρP

(3)

in which K0 is the apparent equilibrium constant, (ρiαi) is the number density of free species i, ρP is the pair number density, gHS c is the contact hard sphere radial distribution function (given in ref 4 and 5), and zi′ ≡ (zi − ησi 2)/(1 + Γσi)

(4)

λ ≡ βe 2 /4πεε0

(5)

σP ≡ σ+ + σ −

(6)

with η and Γ the classical MSA parameters,1−6 β = 1/kT, ε0 is the permittivity of a vacuum, and ε is the relative permittivity of solution. Expressions for η and Γ are given in eqs 7−11 of ref 7. In eq 3, λ0 and σP,0 correspond to λ and σP for an infinitely diluted solution, respectively. The ion pair is constituted by two ions being in contact. In this work, the size of an ion is the same as to whether it is free or it belongs to a pair. Thus, this model for association is best-suited to the description of ion pairing in which in particular the cation keeps its hydration shell. It would not be adapted to the association of acids.8 The excess MM mean activity coefficient can be obtained simply from the thermodynamic formula ln y±MM = Δln y±HS +

β ΔABiMSA + ΔΦBiMSA ρt

(7)

and the total osmotic coefficient is Φ MM = 1 + ΔΦHS + ΔΦBiMSA

(8)

in which ΔA and ΔΦ are the BiMSA contributions to the Helmholtz energy per volume unit and the osmotic coefficient (eqs 5−18 in ref 9), and ρt is the total number HS density, ρt ≡ ρ++ ρ−. Besides, Δln γHS are the hard ± and ΔΦ sphere contributions to the mean ionic activity and osmotic coefficients, respectively. They can be calculated from the Boublik−Mansoori−Carnahan−Starling−Leland expression, as described in ref 4. In this work, as in refs 4−10, the anion size is kept constant (equal to its crystallographic value or adjusted so as to yield an optimum fit), and the diameter of cation, σ+, and the relative permittivity of solution, ε−1, are chosen as linear functions of the salt concentration, Cs, as BiMSA

3. CALCULATIONS The relevant equations have been given previously.3−6,9 For clarity, we will recall only the main relations. The BiMSA model provides an expression for the excess Helmholtz energy of solution per volume unit (ΔABiMSA) at the Mac-Millan Mayer (MM) level, on a molarity scale. It consists of two terms: an electrostatic term (el) related to the presence of ions of charge z+ and z− and of diameters σ+ and σ− and a mass action law term (MAL), originating from association between two ions of opposite sign (see ref 3 for details). The total Helmholtz energy moreover contains a hard-sphere term (HS) related to the effect of excluded volume. Thus, ΔA is written as follows ΔA = ΔAHS + ΔABiMSA

(2)

σ+ = σ+(0) + σs(1)Cs ε−1 = εw −1(1 + αCs)

(1) 9662

BiMSA

(9) (10)

dx.doi.org/10.1021/ie202954y | Ind. Eng. Chem. Res. 2012, 51, 9661−9668


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with εW as the relative permittivity of pure solvent (εW = 78.3 for water). Equations 9 and 10 have been used in several papers dealing with MSA and BiMSA5−7 and are used here to provide a set of parameters compatible with those previously obtained. In eq 9, σ(0) + is the diameter of cation at infinite dilution. The parameter σ(1) S accounts for the variation of cation diameter with concentration. Equation 9 may be viewed as a Taylor expansion of σ+ in powers of the concentration because its variation is generally relatively small.5 Equation 10 is motivated by the fact that experimental relative permittivities seem to often follow this dependency.5 The parameter α accounts for the variation of the permittivity of solution with concentration. It should be noticed that the relative permittivity ε of eq 10 is not expected to coincide with the experimental value because the latter is a macroscopic quantity describing the interactions between ions at large separation while the former is introduced to represent ion−ion electrostatic interactions for short separations.13 In order to compare calculated and experimental osmotic coefficients, the values obtained from the BiMSA model need be converted from the Mac-Millan to the Lewis−Randall (LR) framework (MM-to-LR conversion). To that end, simple expressions reported previously have been used.5 For one electrolyte in water, the mean ionic and osmotic coefficients can be written as

the literature.14−25 For monomethyl-, dimethyl-, and trimethylammonium salts, no values are available in the literature. For simplicity reasons, these densities were interpolated as follows between those for the corresponding ammonium and tetramethylammonium salts with the same anion. For a salt containing a polymethylammonium cation, NH4−nMen+X− (with X− the anion), the density was computed by using the formula dNH4−nMen+X − =

(11) (12)

with Ct =

4

(16)

Density values measured for aqueous solutions of NMe3H+Cl− agreed within 1% with the values obtained using eq 16. The densities of methylsulfonic (HCH3SO3) and ethylsulfonic (HC2H5SO3) acid, sodium methanesulfonate (NaCH3SO3), and ethanesulfonate (NaC2H5SO3) have been measured (see the Experimental section). Densities for solutions containing trifluoromethanesulfonate anions were taken for refs 26 and 27. For all solutions reported here, the values of the parameters a and b appearing in eq 14 are reported in Table 1 of the Supporting Information file. 4.2. Salts Containing Ammonium Cations (Set S1). For the first set of salts, S1, osmotic coefficients reported in the literature11,28−33 are described over the whole concentration range available. Results are collected in Table 2 of the Supporting Information. Plots of the osmotic coefficients as a function of concentration for chloride and nitrate salts are presented in Figures 1 and 2. Discussion of the values for the adjusted parameters and of the absolute average relative deviation obtained from the fitting procedures will be expanded hereafter.

⎛ d ⎞ ln γ±LR = ln y±MM − ln⎜(1 + msMs) w ⎟ − C tV±φ MM ⎝ d ⎠ φLR = φ MM(1 − C tV±)

(4 − n)dNH4+X − + ndNMe4+X−

νmsd 1 + msMs

as the total solute concentration. In these relations, γLR and ϕLR stand for the experimental mean ionic activity coefficient and osmotic coefficients respectively, on molality scale, yMM is the mean ionic activity coefficient as defined in eq 7, and ν, ms, and Ms stand for the total stoichiometric number of ions, the molality of the salt, and the molecular weight of the salt, respectively. Here dw is the density of water. The mean ionic molar volume V± can be calculated from V± =

(Ms − d′)(1 + msMs) νd(1 + msMs − msd′)

(13)

with d as the density of solution and d′ as the derivative of density with respect to salt concentration.

d′ =

∂d ∂Cs

Figure 1. Plots of experimental and calculated osmotic coefficients at 25 °C for aqueous solutions of polyalkylammonium chloride salts: (right-pointing triangle) NH3Me+Cl−; (down-pointing triangle) NH2Me2+Cl−; (triangle) NHMe3+Cl−; (circle) NMe4+Cl−; (square) NEt4+Cl−; (times) NPr4+Cl−; (solid line) calculated values.

(14)

4. RESULTS AND DISCUSSION 4.1. Densities. As in previous work,5,6 the following expression for solution density was used d = d w + aCs − bCs 3/2

Tetraalkylammonium Salts. A first approach of the description of aqueous solutions of tetraalkylammonium halide salts up to a concentration of 6 mol·kg−1 within the BiMSA model was presented earlier by our group.6 It was found that the values for the association constants and the cation diameters were not satisfactory. For instance, a diameter of

(15)

with dW = 997.05 kg·m−3, a and b two parameters to be adjusted, and Cs is in moles per cubic decimeter. Densities for solutions containing cations tetraalkylammonium and chloride, bromide, or iodide anions were taken from 9663



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from 5.54 to 8.80 Å in the series NMe4+, NEt4+, NPr4+. It should be noticed that the values for the infinite dilution diameters σ(0) + , obtained for the tetraalkylammonium cations are very close to values reported earlier in the literature. For instance, the diameter of the tetramethylammonium cation calculated from the ionic volumes available in the literature (5.6035 or 5.5836 Å) is close to the one obtained using the BiMSA model (5.54 Å). Similarly, values of 6.7435 or 6.6836 Å for the tetraethylammonium cation compare well with the value of 6.65 Å obtained from the present model. The value of 8.80 Å obtained for the tetrapropylammonium cation is somehow higher than that reported by Marcus (7.58 Å). This might be due to some limitations of the BiMSA model which describes cations containing long and flexible alkyl chains as charged hard spheres. Previously, the MSA cation diameter was considered to include a solvation sphere that could vary in size as a function of salt concentration. This is reflected in the introduction of a concentration dependent parameter for this diameter, σ(1) s , as defined in eq 8. In the case of tetraalkylammonium cations, σ(1) s can reflect two other phenomenon. First, the carbons of the alkyl chain can rotate freely, so confering some flexibility to the alkyl chain. In order to reduce steric hindrance, contraction of the cation might occur when concentration is increased. Second, the tetrahedral geometry of the tetraalkylammonium cation makes it possible for the anion to penetrate between the alkyl arms of the cation. Because the anion size is kept constant within the BiMSA model, this would result in a decrease in cation size. The results obtained here support this view. All σ(1) s values are negative. Besides, the concentration dependence for the diameter of tetrapropylammonium cation, which contains long and flexible alkyl chains, is significantly higher that those obtained for the other tetramethyl- or tetraethylammonium cations. Because, basically, there are fewer water molecules and therefore solvent dipoles per unit volume when the salt concentration is increased, the permittivity of a solution is expected to decrease with salt concentration.13 This fact must result in a positive value of α (see eq 9). Furthermore, for a given ion (cation or anion), the bigger the counterion, the lower the number of water molecules per unit volume. One thus expects the value of α to increase with the size of the counterion. Experimental values for the permittivity of tetramethylammonium and tetraethylammonium bromide solutions up to 4 mol·kg−1 have been reported previously.37 Let us notice that even though it is not the intention of this work to describe experimental permittivities,13 the values obtained within the BiMSA model are in fair agreement with the permittivity data of some aqueous solutions, such as those containing NMe4Br and NEt4Br (see Figure 5).37 Experimental and calculated values do not coincide for other solutions for which permittivities exhibit unexpected behavior. The experimental values for ε of NPr4Br, for instance, decreases between 0 and 2 mol·kg−1, and then increases sharply to reach a value of 81 at 3.6 mol·kg−1.37 The values for the association constant obtained here for all tetraalkylammonium salt solutions are fairly low in most cases. The greater the ion size, the larger the association constant. This may be explained by the formation of ion pairs facilitated by the increase in hydrophobicity of the cation resulting form the presence of alkyl chains. Because the accuracy of the

Figure 2. Plot of experimental and calculated osmotic coefficients at 25 °C for aqueous solutions of polyalkylammonium nitrate salts: (right-pointing triangle) NH3Me+NO3−; (down-pointing triangle) NH2Me2+NO3−; (triangle) NHMe3+NO3−; (circle) NMe4+NO3−; (square) NEt4+NO3−; (solid line) calculated values.

1.86 Å was regressed for the tetramethylammonium cation (NMe4+), which is much smaller than the value obtained for the ammonium cation (3.3 Å). Besides, the NMe4Cl salt solution was treated as being completely dissociated which is apparently not consistent with experimental results.34 The latter pointed to a (low) value of ca. 1 L·mol−1 for the association constant. In this paper, more realistic parameter values have been obtained by describing aqueous tetraalkylammonium solutions over the whole concentration range available. The systems containing tetramethylammonium (NMe4+), tetraethylammonium (NEt4+), and tetrapropylammonium (NPr4+) cations were reconsidered starting from the following observations. All tetraalkylammonium cations should exhibit BiMSA diameter values higher than that assigned to the ammonium cation. Besides, considering that tetraalkylammonium cations are “hydrophobic” cations, their diameter values should be close to the “crystallographic” values. Therefore, all systems containing a given cation were fitted at the same time, with values for the cation diameter bound in the range from 3.4 to 9 Å. The value for the cation diameter at infinite dilution leading to the smallest overall average absolute relative deviation (AARD) for all salts containing the cation was retained. The results are shown in Table 2 of the Supporting Information file, and osmotic coefficients for tetraalkylammonium chloride and nitrate solutions are plotted in Figures 1 and 2. It is observed that the results obtained using this procedure are satisfactory. The overall AARDs reported are low, with values of the cation diameter that are larger than previously found.6 Experimental and calculated osmotic coefficients for NMe4Cl plotted in Figure 1 exhibit some discrepancies above 8 mol·kg−1. Generally, AARD values are higher for solutions containing NEt4+ or NPr4+ cations. A detailed discussion of the parameter values obtained in this study will now be given. First, the values obtained in this work for the infinite dilution diameters of tetraalkylammonium cations are all greater than those obtained previously for the ammonium cation (3.40 Å).5 It is moreover satisfying to observe that the values increase with the number of methyl groups appended to the ammonium, 9664



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In agreement with the fact that the permittivity should decrease with the size of cation, α increases slightly with the number of methyl groups (for a given anion). Finally, for a given anion, the association constants regressed for polymethylammonium solutions are always comprised between those obtained for the corresponding ammonium and tetramethylammonium solutions. As shown in Figure 4, K increases monotonically with the number of methyl groups on the nitrogen atom, in the case of chlorides, nitrates, and perchlorates. Overall, these results show that the BiMSA model is able to describe the thermodynamic properties of solutions made up of bulky and even asymmetrical ammonium cations. 4.3. Salts Containing Polyatomic Anions (Set S2). Osmotic coefficients reported in the literature for aqueous solutions containing bulky and polyatomic anion salts were described over the whole concentration range available.39−45 Average absolute relative deviation and adjusted parameter values are collected in Table 3 of the Supporting Information file. Plots of the osmotic coefficients for several sodium salts described here are presented in Figure 3.

measurements of such low association constants is questionable,38 a thorough discussion of our values is not possible. Nevertheless, the values obtained are close to the experimental ones, when available. For instance, a value of 1.03 L·mol−1 is found for NMe4Cl as compared to 1 L·mol−1 reported in the literature.34 Besides, the experimental values for the association constants of NMe4Br (3.0), NEt4Br (3.7), and NPr4Br (2.8), obtained by dielectric relaxation spectroscopy,37 are in reasonable agreement with our adjusted values of ca. 1.88, 1.94, and 3.33, respectively. In order to test the ability of the BiMSA model to account for unusual ions, the osmotic coefficients of solutions containing the ethylenebis(trimethylammonium) cation and the chloride, iodide, or sulfate anion are described. This doubly charged cation is equivalent to two tetramethylammonium cations bound together. Because osmotic coefficients were reported for relatively low concentrations (below 4 mol·kg−1), because no density data were available, and because these three systems have been studied only to test the ability of the BiMSA model to describe solutions containing unsymmetrical ions, the densities of these solutions were approximated by that of pure water. The optimized diameter found for this cation is 8.53 Å. It is less than twice the diameter of a tetramethylammonium cation. Surprisingly, it is close to the value obtained for the tetrapropylammonium cation. Despite the fact that this cation is not spherical, the BiMSA model accurately describes the solutions of corresponding chloride, iodide, and sulfate salts. AARD values are found to be acceptable, with plausible parameter values. In particular, the K values range between 15 and 30 L·mol−1, which are significantly higher that those obtained for other ammonium salts. Polymethylammonium Salt Solutions. The osmotic coefficients for nine electrolyte solutions containing mono-, di-, or trimethylammonium cations and the chloride, nitrate, or perchlorate anion have been described. The same fitting procedure as that used for tetraalkylammonium salt solutions was employed, with refined values for the cation diameters ranging from 3.30 to 5.6 Å. For each cation, aqueous solutions containing anions chloride, nitrate, or perchlorate were fitted in the same run and the final value adopted for the cation diameter corresponded to the lowest observed overall AARD. For all nine systems, the results were found to be satisfying. AARD values are low, typically below 0.4%. Moreover, the value for the cation diameter at infinite dilution, σ(0) + , increases with the number of methyl groups appended on the nitrogen atom. Regarding the monomethylammonium cation as spherical leads to a volume of 39 Å3 as calculated from its diameter within our model, which turns out to be very close to the “crystallographic” volume of 41 Å3 reported elsewhere.36 The parameter σ(1) S , on the other hand, decreases with the number of methyl groups, whatever the anion. The values obtained for NHMe3Cl and NHMe3NO3 are close to, yet slightly higher than, those obtained for NMe 4Cl and NMe4NO3, respectively. Furthermore, σS(1) values are always ranging between those obtained for the corresponding ammonium and tetramethylammonium salts. As in the case of the NH4+ cation, σS(1) for polymethylammonium cations reflects the variation of the solvation sphere with concentration. Therefore, the higher the number of hydrophobic methyl substituents, the smaller the solvation sphere and the lower the value for σ(1) S .

Figure 3. Plot of the experimental and calculated osmotic coefficients at 25 °C for aqueous solutions of sodium salts: (square) Na+CH3SO4−; (down-pointing triangle) Na+CH3SO3−; (circle) Na+CF3SO3−; (triangle) Na+CF3CO2−; (solid line): calculated values. (inset) Zoom in between 0 and 3 mol·kg−1.

In this section, all salts studied here have been described using cation diameters optimized in previous works.5 The results are satisfactory, with low AARD’s in all cases. As shown in Figure 4, even for systems containing sodium cations and very different osmotic coefficient values, the BiMSA model succeeds in representing the experimental data, with plausible parameter values. Osmotic coefficients for solutions containing NaCH3SO4 have been reported up to 19 mol·kg−1, which is the highest concentration for the solutions studied in this section. Besides, the curve for the osmotic coefficient reaches a sharp maximum for a concentration of ca. 9 mol·kg−1. The osmotic coefficient is well described at high molality values, but as shown in the inset in Figure 4, discrepancies exist below 3 mol·kg−1. The BiMSA parameters will now be discussed. Only the mean ionic activity coefficients of sodium tetrafluoroborate aqueous solutions have been reported in the literature.42 Besides, a value of 3.64 Å for the tetrafluoroborate anion was reported by Marcus et al.35 In the present study, this 9665



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than that of ethanesulfonate. Similarly to the case of acetates, CF3SO3− is the largest of the sulfonate anions. It is interesting to note that its diameter value adjusted here (6.16 Å) is close to a value of 6.14 Å obtained previously from conductivity measurements,47 despite the fact that such measurements are not reliable for obtaining anion radii.48 One observes that, with the same alkyl chain, sulfonate anions are systematically smaller than acetates. The anion diameter found for the methylsulfate, on the other hand, is smaller than for all other anions studied in this work. Because a methylsulfate anion consists of one CH3 group and one SO3− group bound together by an oxygen atom and because a methanesulfonate anion corresponds to one CH3 group and one SO3− group directly linked together, one would expect the methylsulfate anion to be larger than methanesulfonate or even acetate. A tentative explanation of this result is that the charge density on alkanesulfonate anions is expected to be more localized than on the alkylsulfate moieties. Then, one may expect the latter anions to be less hydrated, and therefore smaller, than alkanesulfonate ones. The values of σ(1) S for most salts studied in this section are particularly high. Such results have been previously mentioned for acetate solutions.6 This might be due to the fact that these anions are significantly hydrated, as suggested by the values of their diameters obtained here. Such an effect, which is not taken into account in this study, will require more attention in future work. HCF3SO3 and HCH3SO3 are commonly regarded as fully dissociated acids. In the present model, HCH3SO3 appears to be fully dissociated and HCF3SO3 exhibits a low association constant (K = 0.10). The value of 0.46 obtained for the association constant of HC2H5SO3 is most probably due to the presence of an ethyl group. As observed for polyalkylammonium cation, the association constant is expected to increase with the alkyl chain length. Surprisingly, even though all sodium salts exhibit association constant ranging between 0.16 and 0.76, NaC2H5SO3 appears to be fully dissociated. No clear explanation can be given for this behavior. Note that the data reported in the literature for this salt are activity coefficients calculated from experimental values of the osmotic coefficient which are not tabulated in ref 40.

Figure 4. Association constant of aqueous solutions containing polyalkylammonium cations: (square) polyalkylammonium chloride salts; (triangle) polyalkylammonium nitrate salts; (circle) polyalkylammonium perchlorate salts; (diamond) tetraalkylammonium bromide salts.

Figure 5. Plot of experimental and calculated relative permittivities for aqueous solutions of tetraalkylammonium bromide at 25 °C: (circle) tetramethylammonium bromide; (square) tetraethylammonium bromide; (solid line): calculated values. Experimental data taken from ref 37.

salt is well described with an AARD of 0.44% and an optimized anion diameter of 3.22 Å. In this work, the optimized anion diameter for CF3CO2− is 6.46 Å. Because of the presence of the trifluoromethyl group, it is satisfying to notice that this value is larger than that of CH3CO2− previously reported (5.95 Å).6 CF3CO2− is the largest reported in this study. Note that a value of 3.80 Å for the acetate anion had been calculated by Marcus using the Born theory and experimental measurements of the free enthalpy of solvation.35 A recent paper reported hydration numbers of ca. 5 for formate and acetate anions, suggesting a significant hydration of these anions in water.46 Moreover, the hydration number for CH3CO2− appeared to be above that for HCO2−, which suggests a significant hydration of the acetate anion. The large sizes obtained for CH3CO2− and CF3CO2− within the BiMSA model appear to be in agreement with these recent results. A value of 4.4 Å was previously optimized for the sulfate ion.5 No diameter values seem to be available in the literature for alkanesulfonate ions. The diameters adjusted for the alkanesulfonate anions seem plausible. First, they are larger than the sulfate ion. Furthermore, the size of methanesulfonate is smaller

5. CONCLUSION The BiMSA model is capable of representing the deviations from ideality in solutions of alkylammonium cations and bulky anions. A set of adjusted parameter values has been determined. Cation sizes are plausible and increase with the number and the length of the alkyl chains. Association constants are of the same order of magnitude as those reported in the literature, when available. As expected from basic physical considerations, the values of the regressed cation diameter and relative permittivity decrease with salt concentration. The BiMSA model also accurately describes solutions of a highly asymmetric cation, namely ethylenebis(trimethylammonium) cations. The infinite dilution size of this doubly charged cation is comprised between those of tetraethyl- and tetrapropylammonium. The association constants appear to be higher than for the polyalkylammonium salts. Aqueous solutions of bulky anions salts have also been accurately represented within the BiMSA model. Optimized 9666



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(10) Torres-Arenas, J.; Simonin, J. P.; Bernard, O.; Ruas, A.; Moisy, P. Thermodynamics of binary and ternary solutions of multivalent electrolytes with formation of 1:1 and 1:2 complexes, within the mean spherical approximation. Ind. Eng. Chem. Res. 2010, 49, 1937. (11) Lindenbaum, S.; Boyd, G. E. Osmotic and activity coefficients for the symmetrical tetraalkylammonium halides in aqueous solution at 25°. J. Phys. Chem. 1964, 68, 911. (12) Perron, G.; Desnoyers, J. E. Volumes and heat capacities of sodium perfluoroalkanoates in water. J. Chem. Eng. Data 1997, 42, 172. (13) Simonin, J. P.; Bernard, O.; Blum, L. Description of the thermodynamic properties of aqueous ionic solutions within the Mean Spherical Approximation. Oil Gas Sci. Tech. 2008, 63, 321. (14) Gerchikova, M. N.; Orekhov, I. I.; Kopylov, E. A.; Karavan, S. V. Densities of aqueous solutions of tetramethylammonium, tetraethylammonium and (2-hydroxyethyl)trimethylammonium chlorides. J. Gen. Chem. USSR 1978, 48, 214. (15) Timmermanns, J. The physico-chemical constants of binary systems in concentrated solutions; Interscience: New York; 1960. (16) Wen, W. Y.; Saito, S. Apparent and partial molal volumes of five symmetrical tetraalkylammonium bromides in aqueous solutions. J. Phys. Chem. 1964, 68, 2639. (17) Hückel, E.; Schaaf, H. Experimentelle untersuchungen über die konzentrations- und temperaturabhängigkeit der zähigkeit wäßriger lösungen starker elektrolyte. Z. Phys. Chem. 1959, 21, 326. (18) Levien, B. J. Some physical properties of aqueous solutions of tetramethylammonium bromide and tetramethylammonium iodide. Aust. J. Chem. 1965, 18, 1161. (19) Soffer, N.; Bloemendal, M.; Marcus, Y. Molar refractivities of tetra-n-alkylammonium salts and ions. J. Chem. Eng. Data 1988, 33, 43. (20) Lowe, B. M.; Rubienski, G. A. Viscosity B-coefficients for triethyl-n-propylammonium iodide, ethyltri-n-propylammonium iodide and tetra-n-propylammonium iodide in aqueous solution at 25°C. Electrochim. Acta 1974, 19, 393. (21) Ryazanov, M. A. Determination of the density of aqueous solutions of tetramethylammonium nitrate at 298.15 K. Zh. Pirkladnol. Khimii 1991, 65, 1183. (22) Novotny, P.; Söhnel, O. Densities of binary aqueous solutions of 306 inorganic substances. J. Chem. Eng. Data 1988, 33, 49. (23) Crooks, W. J.; Christian, J. D. Densities and partial molal volumes of sodium tetrafluoroborate aqueous solutions at 20°C. Indian J. Chem. 2004, 43A, 1872. (24) Koda, S.; Nomura, H. Aqueous solutions of sodium methylsulfate by Raman scattering, NMR, ultrasound, and density measurements. J. Solution Chem. 1985, 14, 355. (25) Dawson, L. R.; Leader, G. R.; Zimmerman, H. K. The conductivity of concentrated electrolyte systems as a function of temperature and concentration. J. Phys. Chem. 1951, 55, 447. (26) Xiao, C.; Tremaine, P. R. Apparent molar volumes of aqueous sodium trifluoromethanesulfonate and trifluoromethanesulfonic acid from 283 to 600 K and pressures up to 20 MPa. J. Solution Chem. 1997, 26, 277. (27) Xiao, C.; Pham, T.; Xie, W.; Tremaine, P. R. Apparent molar volumes and heat capacities of aqueous trifluoromethanesulfonic acid and its sodium salt from 283 to 328 K. J. Solution Chem. 2001, 30, 201. (28) Kalman, E.; Schwabe, K. Osmotic and activity coefficients of ammonium thiocyanate in aqueous solutions at 25 °C. J. Solution Chem. 1979, 8, 1. (29) Macaskill, J. B.; Bates, R. G. Osmotic and activity coefficients of monomethyl-, dimethyl-, and trimethylammonium chlorides at 25°C. J. Solution Chem. 1986, 15, 323. (30) Bonner, O. D. Osmotic and activity coefficients of methylsubstituted ammonium nitrates at 298.15 K. J. Chem. Eng. Data 1981, 26, 148. (31) Bonner, O. D. Osmotic and activity coefficients of methylsubstituted ammonium perchlorates at 298.15 K. J. Chem. Eng. Data 1982, 27, 62. (32) Bonner, O. D. The osmotic and activity coefficients of some salts having relatively large molar volumes. J. Chem. Eng. Data 1976, 21, 498.

diameters for bulky anions have been obtained. In all cases, the values adjusted are realistic. The K values are in agreement with previous observations and reported values, when available. The results are at least of the same accuracy as those reported previously in the literature using the Pitzer and the eNRTL model.49 On the other hand, as compared to these models, the BiMSA parameters (ion sizes and solution permittivity) have some physical meaning. A consequence of this feature is that individual sizes at infinite dilution are assigned to cations and to anions (see Tables 2 and 3 of the Supporting Information). These results are very encouraging. They pave the way for a future description of aqueous solutions of ionic liquids, because the latter are in a large majority composed with quaternary cyclic ammonium cations, such as imidazolium, and anions such as halide, tetrafluoroborate, or alkylsulfonate. The BiMSA model therefore represents an interesting alternative to semiempirical thermodynamic models.

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ASSOCIATED CONTENT

S Supporting Information *

Tables 1−3 collecting the parameters used for eq 16 and all BiMSA parameters adjusted in this work for the description of aqueous solutions containing alkylammonium cations or bulky anions. This information is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 00 33 479758838. Notes

The authors declare no competing financial interest.

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REFERENCES

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Solutions of Alkylammonium and Bulky Anions: Description of Osmotic

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